Integral limit theorems for sums of additive functions with shifted arguments

In this paper we find necessary and sufficient conditions for the vanishing of the limit distribution for a linear combination of two real-valued additive functions. We obtain results for $g_1(an+b)/B_1(x)+g_2(cn+d)/B_2(x)-A(x)$, where $a>0$, $b,c>0$, and $d$ are integers with $ad-bc\ne 0$, that are almost as strong as in the case of a single additive function. As an application, we resolve a conjecture of Katai.